# Inference¶

## Two-graph hypothesis testing¶

class graspy.inference.LatentPositionTest(embedding='ase', n_components=None, n_bootstraps=500, test_case='rotation')[source]

Two-sample hypothesis test for the problem of determining whether two random dot product graphs have the same latent positions.

This test assumes that the two input graphs are vertex aligned, that is, there is a known mapping between vertices in the two graphs and the input graphs have their vertices sorted in the same order. Currently, the function only supports undirected graphs.

Parameters: embedding : string, { 'ase' (default), 'omnibus'} String describing the embedding method to use: 'ase' Embed each graph separately using adjacency spectral embedding and use Procrustes to align the embeddings. 'omnibus' Embed all graphs simultaneously using omnibus embedding. n_components : None (default), or int Number of embedding dimensions. If None, the optimal embedding dimensions are found by the Zhu and Godsi algorithm. test_case : string, {'rotation' (default), 'scalar-rotation', 'diagonal-rotation'} describes the exact form of the hypothesis to test when using 'ase' or 'lse' as an embedding method. Ignored if using 'omnibus'. Given two latent positions, $$X_1$$ and $$X_2$$, and an orthogonal rotation matrix $$R$$ that minimizes $$||X_1 - X_2 R||_F$$: 'rotation' $H_o: X_1 = X_2 R$ 'scalar-rotation' $H_o: X_1 = c X_2 R$ where $$c$$ is a scalar, $$c > 0$$ 'diagonal-rotation' $H_o: X_1 = D X_2 R$ where $$D$$ is an arbitrary diagonal matrix n_bootstraps : int, optional (default 500) Number of bootstrap simulations to run to generate the null distribution null_distribution_1_, null_distribution_2_ : np.ndarray (n_bootstraps,) The distribution of T statistics generated under the null, using the first and and second input graph, respectively. The latent positions of each sample graph are used independently to sample random dot product graphs, so two null distributions are generated sample_T_statistic_ : float The observed difference between the embedded positions of the two input graphs after an alignment (the type of alignment depends on test_case) p_value_1_, p_value_2_ : float The p value estimated from the null distributions from sample 1 and sample 2. p_value_ : float The overall p value from the test; this is the max of p_value_1_ and p_value_2_

References

 [R3d27477db6c1-1] Tang, M., A. Athreya, D. Sussman, V. Lyzinski, Y. Park, Priebe, C.E. "A Semiparametric Two-Sample Hypothesis Testing Problem for Random Graphs" Journal of Computational and Graphical Statistics, Vol. 26(2), 2017
fit(self, A1, A2)[source]

Fits the test to the two input graphs

Parameters: A1, A2 : nx.Graph, nx.DiGraph, nx.MultiDiGraph, nx.MultiGraph, np.ndarray The two graphs to run a hypothesis test on. If np.ndarray, shape must be (n_vertices, n_vertices) for both graphs, where n_vertices is the same for both self
fit_predict(self, A1, A2)

Fits the model and returns the p-value Parameters ---------- A1, A2 : nx.Graph, nx.DiGraph, nx.MultiDiGraph, nx.MultiGraph, np.ndarray

The two graphs to run a hypothesis test on. If np.ndarray, shape must be (n_vertices, n_vertices) for both graphs, where n_vertices is the same for both
p_value_ : float
The overall p value from the test
get_params(self, deep=True)

Get parameters for this estimator.

Parameters: deep : bool, default=True If True, will return the parameters for this estimator and contained subobjects that are estimators. params : mapping of string to any Parameter names mapped to their values.
set_params(self, **params)

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as pipelines). The latter have parameters of the form <component>__<parameter> so that it's possible to update each component of a nested object.

Parameters: **params : dict Estimator parameters. self : object Estimator instance.
class graspy.inference.LatentDistributionTest(test='dcorr', metric='euclidean', n_components=None, n_bootstraps=200, workers=1, size_correction=True, pooled=False)[source]

Two-sample hypothesis test for the problem of determining whether two random dot product graphs have the same distributions of latent positions.

This test can operate on two graphs where there is no known matching between the vertices of the two graphs, or even when the number of vertices is different. Currently, testing is only supported for undirected graphs.

Parameters: test : str (default="hsic") Backend hypothesis test to use, one of ["cca", "dcorr", "hhg", "rv", "hsic", "mgc"]. These tests are typically used for independence testing, but here they are used for a two-sample hypothesis test on the latent positions of two graphs. See hyppo.ksample.KSample for more information. metric : str or function (default="gaussian") Distance or a kernel metric to use, either a callable or a valid string. If a callable, then it should behave similarly to either sklearn.metrics.pairwise_distances() or to sklearn.metrics.pairwise.pairwise_kernels(). If a string, then it should be either one of the keys in either sklearn.metrics.pairwise.PAIRED_DISTANCES or in sklearn.metrics.pairwise.PAIRWISE_KERNEL_FUNCTIONS, or "gaussian", which will use a gaussian kernel with an adaptively selected bandwidth. It is recommended to use kernels (e.g. "gaussian") with kernel-based hsic test and distances (e.g. "euclidean") with all other tests. n_components : int or None (default=None) Number of embedding dimensions. If None, the optimal embedding dimensions are found by the Zhu and Godsi algorithm. See selectSVD() for more information. n_bootstraps : int (default=200) Number of bootstrap iterations for the backend hypothesis test. See hyppo.ksample.KSample for more information. workers : int (default=1) Number of workers to use. If more than 1, parallelizes the code. Supply -1 to use all cores available to the Process. size_correction: bool (default=True) Ignored when the two graphs have the same number of vertices. The test degrades in validity as the number of vertices of the two graphs diverge from each other, unless a correction is performed. If True, when the two graphs have different numbers of vertices, estimates the plug-in estimator for the variance and uses it to correct the embedding of the larger graph. If False, does not perform any modifications (not recommended). pooled: bool (default=False) Ignored whenever the two graphs have the same number of vertices or size_correction is set to False. In order to correct the adjacency spectral embedding used in the test, it is needed to estimate the variance for each of the latent position estimates in the larger graph, which requires to compute different sample moments. These moments can be computed either over the larger graph (False), or over both graphs (True). Setting it to True should not affect the behavior of the test under the null hypothesis, but it is not clear whether it has more power or less power under which alternatives. Generally not recomended, as it is untested and included for experimental purposes. null_distribution_ : ndarray, shape (n_bootstraps, ) The distribution of T statistics generated under the null. sample_T_statistic_ : float The observed difference between the embedded latent positions of the two input graphs. p_value_ : float The overall p value from the test.

References

 [R63152004fa12-1] Tang, M., Athreya, A., Sussman, D. L., Lyzinski, V., & Priebe, C. E. (2017). "A nonparametric two-sample hypothesis testing problem for random graphs." Bernoulli, 23(3), 1599-1630.
 [R63152004fa12-2] Panda, S., Palaniappan, S., Xiong, J., Bridgeford, E., Mehta, R., Shen, C., & Vogelstein, J. (2019). "hyppo: A Comprehensive Multivariate Hypothesis Testing Python Package." arXiv:1907.02088.
 [R63152004fa12-3] Varjavand, B., Arroyo, J., Tang, M., Priebe, C., and Vogelstein, J. (2019). "Improving Power of 2-Sample Random Graph Tests with Applications in Connectomics" arXiv:1911.02741
 [R63152004fa12-4] Alyakin, A., Agterberg, J., Helm, H., Priebe, C. (2020) "Correcting a Nonparametric Two-sample Graph Hypothesis Test for Graphs with Different Numbers of Vertices" TODO cite the arXiv whenever possible
fit(self, A1, A2)[source]

Fits the test to the two input graphs

Parameters: A1, A2 : nx.Graph, nx.DiGraph, nx.MultiDiGraph, nx.MultiGraph, np.ndarray The two graphs to run a hypothesis test on. self
fit_predict(self, A1, A2)

Fits the model and returns the p-value Parameters ---------- A1, A2 : nx.Graph, nx.DiGraph, nx.MultiDiGraph, nx.MultiGraph, np.ndarray

The two graphs to run a hypothesis test on. If np.ndarray, shape must be (n_vertices, n_vertices) for both graphs, where n_vertices is the same for both
p_value_ : float
The overall p value from the test
get_params(self, deep=True)

Get parameters for this estimator.

Parameters: deep : bool, default=True If True, will return the parameters for this estimator and contained subobjects that are estimators. params : mapping of string to any Parameter names mapped to their values.
set_params(self, **params)

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as pipelines). The latter have parameters of the form <component>__<parameter> so that it's possible to update each component of a nested object.

Parameters: **params : dict Estimator parameters. self : object Estimator instance.